# Global analysis of multi-strains SIS, SIR and MSIR epidemic models

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1 MASAIE - Tools and models of nonlinear control theory for epidemiology and immunology
LMAM - Laboratoire de Mathématiques et Applications de Metz, Inria Nancy - Grand Est, IECL - Institut Élie Cartan de Lorraine
Abstract : We consider SIS, SIR and MSIR models with standard mass action and varying population, with $n$ different pathogen strains of an infectious disease. We also consider the same models with vertical transmission. We prove that under generic conditions a competitive exclusion principle holds. To each strain a basic reproduction ratio can be associated. It corresponds to the case where only this strain exists. The basic reproduction ratio of the complete system is the maximum of each individual basic reproduction ratio. Actually we also define an equivalent threshold for each strain. The winner of the competition is the strain with the maximum threshold. It turns out that this strain is the most virulent, i.e., this is the strain for which the endemic equilibrium gives the minimum population for the susceptible host population. This can be interpreted as a pessimization principle.
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https://hal.inria.fr/hal-00833057
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Derdei Bichara, Abderrahman Iggidr, Gauthier Sallet. Global analysis of multi-strains SIS, SIR and MSIR epidemic models. Journal of Applied Mathematics and Computing, Springer, 2014, 44 (1-2), pp.273-292. ⟨10.1007/s12190-013-0693-x⟩. ⟨hal-00833057⟩

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